(1):

(v. t.) To open and cleanse, as cotton, flax, or wool, by means of a willow. See Willow, n., 2.

(2):

(n.) A machine in which cotton or wool is opened and cleansed by the action of long spikes projecting from a drum which revolves within a box studded with similar spikes; - probably so called from having been originally a cylindrical cage made of willow rods, though some derive the term from winnow, as denoting the winnowing, or cleansing, action of the machine. Called also willy, twilly, twilly devil, and devil.

(3):

(n.) Any tree or shrub of the genus Salix, including many species, most of which are characterized often used as an emblem of sorrow, desolation, or desertion. "A wreath of willow to show my forsaken plight." Sir W. Scott. Hence, a lover forsaken by, or having lost, the person beloved, is said to wear the willow.